Abstract
The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codimension-two submanifold with constant normalized null expansion (null mean curvature) must lie in a shear-free (umbilical) null hypersurface. These results are generalized for higher order curvature invariants. In particular, the notion of mixed higher order mean curvature is introduced to highlight the special null geometry of the submanifold. Finally, Alexandrov type theorems are established for spacelike submanifolds with constant mixed higher order mean curvature, which are generalizations of hypersurfaces of constant Weingarten curvature in the Euclidean space.
| Original language | English |
|---|---|
| Pages (from-to) | 249-290 |
| Number of pages | 42 |
| Journal | Journal of Differential Geometry |
| Volume | 105 |
| Issue number | 2 |
| DOIs | |
| Publication status | Published - 2017 Feb |
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All Science Journal Classification (ASJC) codes
- Analysis
- Algebra and Number Theory
- Geometry and Topology
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Minkowski formulae and Alexandrov theorems in space time. / Wang, Mu Tao; Wang, Ye Kai; Zhang, Xiangwen.
In: Journal of Differential Geometry, Vol. 105, No. 2, 02.2017, p. 249-290.Research output: Contribution to journal › Article
TY - JOUR
T1 - Minkowski formulae and Alexandrov theorems in space time
AU - Wang, Mu Tao
AU - Wang, Ye Kai
AU - Zhang, Xiangwen
PY - 2017/2
Y1 - 2017/2
N2 - The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codimension-two submanifold with constant normalized null expansion (null mean curvature) must lie in a shear-free (umbilical) null hypersurface. These results are generalized for higher order curvature invariants. In particular, the notion of mixed higher order mean curvature is introduced to highlight the special null geometry of the submanifold. Finally, Alexandrov type theorems are established for spacelike submanifolds with constant mixed higher order mean curvature, which are generalizations of hypersurfaces of constant Weingarten curvature in the Euclidean space.
AB - The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codimension-two submanifold with constant normalized null expansion (null mean curvature) must lie in a shear-free (umbilical) null hypersurface. These results are generalized for higher order curvature invariants. In particular, the notion of mixed higher order mean curvature is introduced to highlight the special null geometry of the submanifold. Finally, Alexandrov type theorems are established for spacelike submanifolds with constant mixed higher order mean curvature, which are generalizations of hypersurfaces of constant Weingarten curvature in the Euclidean space.
UR - http://www.scopus.com/inward/record.url?scp=85017470070&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85017470070&partnerID=8YFLogxK
U2 - 10.4310/jdg/1486522815
DO - 10.4310/jdg/1486522815
M3 - Article
AN - SCOPUS:85017470070
VL - 105
SP - 249
EP - 290
JO - Journal of Differential Geometry
JF - Journal of Differential Geometry
SN - 0022-040X
IS - 2
ER -